The answer is usually no. The law in Virginia does not transfer liability to the owner of a rental property for a tenant's animal. However, it might be a good idea to require your renters to have get renters' insurance, which is like home owners' insurance, but for tenants. Then, the insurer will take care of paying and defending any claims for injury or death caused by pets at the apartment building or house.
Furthermore, Virginia follows the law commonly referred to as the "one bite rule." This means the dog owner is shielded from civil liability the first time each of his or her animals attacks another individual, if the attack was not intentional or due to negligence. For example, if the owner caused the dog bite by breaking a leash law, then that person could be held legally liable. If the animal was known to have dangerous propensities, the owner can sometimes be held responsible for even the first time it breaks skin.